The function $f\left(x\right)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at $x=\frac{2k}{3}$ then$k=$

We have $f\left(x\right)=\frac{x}{2}+\frac{2}{x}$ 

$\therefore f\text{'}\left(x\right)=\frac{1}{2}-\frac{2}{x^2}$ and $f"\left(x\right)=\frac{4}{x^3}$

Now, $f\text{'}\left(x\right)=0$ or$x^2=4$ or$x=\pm 2\therefore f"\left(x\right)>0$ for $x=2.$

Therefore,$f$ has local minima at $x=2.$